I have a question on a fundamental property of harmonic functions.
Let $\Omega \subset \mathbb{R}^d$ be a domain. We define $H^{1}(\Omega)$ by \begin{align*} H^{1}(\Omega)=\{u \in L^{2}(\Omega,m) \mid |\nabla u| \in L^{2}(\Omega,m)\}, \end{align*} where $\nabla u$ is the distributional derivative of $u$ and $m$ the Lebesgue measure on $\Omega$.
Let $b:\Omega \to \mathbb{R}^d$ is a bounded measurable function on $\Omega$.
Let $f$ be a function on $\Omega$ satisfying the following property: for any compact subset $K$ of $\Omega$ there exists $u \in H^{1}(\Omega)$ such that $f=u$, $m$-a.e. on $K$. We further assume that $f$ satisfies \begin{align*} \int_{\Omega}(\nabla f, \nabla g)\,dm+\int_{\Omega}(b,\nabla f)g\,dm=0 \end{align*} for any $g \in C^{\infty}_{0}(\Omega)$. Namely, $f$ is a distributional harmonic function on $\Omega$.
Can we show that $f \in C^{2}(\Omega)$?
If $b=0$, it follows that $f \in C^{\infty}(\Omega)$ by the Weyl's lemma.